Differential detection of GFSK signal using decision feedback and preamble for carrier, clock and frame synchronization

ABSTRACT

This invention provides a method for bit detection of the GFSK signals at the receiver end. The bit detection is done digitally after the carrier is removed. It employs differential detection incorporated with decision feedback, which uses previous one (so-called on-bit differential detection), or two (so-called two-bit differential detection) bits as correcting information. In addition, synchronization for bit timing and frequency offset resulted from clocks between the transmitter and receiver are also performed with or without preamble as prior information. If preamble is available, the bit timing and frequency-offset bias are estimated from the preamble, which is the case of this present invention. If preamble is not available, this information is estimated directly from the unknown received signals. Once the information about the bit timing and frequency offset is obtained, it is used for the following bit detection.

FIELD OF INVENTION

The present invention is related to communication systems, and moreparticularly, to the digital modem system using GFSK transmissionscheme.

BACKGROUND OF INVENTION

Please refer to FIG. 1, which shows a typical GFSK receiver structure.The antenna 100 receives a GFSK signal 110 propagating through channel,then the RF circuit 120 removes the carrier from the received GFSKsignal 110 and obtains a baseband analog complex signal z(t) 130. Thepurpose of ADC (analog-to-digital converter) 140 is to sample the analogbaseband complex signal 130 to a digital baseband complex signal z_(k,j)150, and the baseband circuit 160 demodulates and processes the digitalbaseband complex signal 150 and obtains the original binary sequenceb(k) 170.

The invention is related to the digital modem system using GFSK schemeto transmit the signal in the baseband. The digital baseband complexsignals are demodulated using differential detection. In other word, thedecision of a bit is based on the phase difference between the currentand its previous received signals. In addition, to demodulate a signalcorrectly also requires certain mechanism for synchronization. Thesynchronization tasks include carrier frequency, phase and symboltiming. Generally, a communication system will provide extra information(so-called preamble or training sequence) to aid the receiver toaccomplish these tasks. This present invention develops algorithms toperform differential detection of GFSK signal using decision feedbackand preamble (training sequence) for frequency, clock and framesynchronization.

GFSK, which employs Gaussian filter for pulse shaping, is an attractivemodulation scheme due to its compact spectrum. However, theInter-Symbol-Interference (ISI) introduced by the Gaussian filter alsodegrades the bit error rate (BER) performance. Various receiverstructures were proposed to improve the BER performance of GFSK owing tothe ISI resulted from Gaussian filter. ABBAS et al (reference b.1)proposed a method using differential detection with decision feedback toovercome the ISI issue. In their original paper, they only dealt withGMSK modulation and assumed that clock and frequency have beensynchronized perfectly.

As to the synchronization issue, Mehian et al (reference b.2) proposed amethod to estimate the symbol timing and frequency offset withouttraining sequence. In their original paper, they only dealt with GMSKmodulation and used conventional differential detection.

This present invention modifies, combines, and extends their works fromGMSK to GFSK. For a given pre-known preamble, this invention estimatesthe frequency offset using the preamble as prior information andestimate the symbol timing using the estimated frequency offset. Oncethe estimated frequency offset and symbol timing are obtained, thisinformation is used to do differential detection incorporating withdecision feedback.

SUMMARY OF INVENTION

A method and a circuit of estimating a binary sequence in a GFSKcommunication system are disclosed in the present invention. First inresponse to a complex baseband signal and a preamble data t(k), obtain afrequency offset estimation. Then obtain a complex digital decimatedsignal by estimating a sampling point. Based on the complex digitaldecimated signal, the frequency offset estimation, using one-bit andtwo-bit differential detection technique to demodulate the complexdigital decimated signal to generate the binary sequence. Finally, inresponsive to the binary sequence and the preamble data t(k), obtain astarting bit of the binary sequence.

BRIEF DESCRIPTION OF DRAWINGS

The present invention will become fully understood from the detaileddescription given herein below with the accompanying drawings, given byway of illustration only and thus not intend to limit the presentinvention.

FIG. 1 illustrates a typical GFSK receiver structure.

FIG. 2 illustrates the block diagram of present invention.

FIG. 3 illustrates the flowchart of the operation of the carriersynchronization circuit.

FIG. 4 illustrates the mechanism of finding N₁, N₂, N₃, N₄.

FIG. 5 illustrates the flowchart of the operation of the clocksynchronization circuit.

FIG. 6 illustrates the flowchart of the operation of the demodulationcircuit.

FIG. 7 illustrates the flowchart of the operation of the framesynchronization circuit.

DETAILED DESCRIPTION OF PRESENT INVENTION

To describe the invention clearly, a number of definition of terms usedherein are given as follows:

The term “symbol” refers to data represented by more than one bit.

The term “preamble” used herein refers to a data string both sender andreceiver agree to use as an information header embedded in thetransmission signal.

FIG. 2 illustrates the block diagram of the present invention. Oneaspect of the present invention is to use the baseband circuit forestimating a binary sequence b(k)={{circumflex over (b)}₀, {circumflexover (b)}₁, {circumflex over (b)}₂, . . . , {circumflex over (b)}_(k−2),{circumflex over (b)}_(b−1), {circumflex over (b)}b_(k), {circumflexover (b)}_(k+1), . . . } in GFSK communication system. In response to acomplex baseband signal z_(k,j) 200 and a preamble data t(k) defined inthe GFSK communication system, a carrier synchronization circuit 210generates a frequency offset estimation {circumflex over (Ω)}_(Δ) 220,wherein z_(k,j) 200 represents a j-th sample of a k-th symbol of areceived data, 0≦j≦N−1, N represents samples per symbol. A clocksynchronization circuit 230 receives the complex digital signal z_(k,j)200 and the frequency offset estimation {circumflex over (Ω)}_(Δ) 220,estimates a sampling point ĵ and generates a complex digital decimatedsignal z_(k,ĵ) 240 by using the sampling point ĵ. A demodulation circuit250 demodulates the complex digital decimated signal z_(k,ĵ) 240 togenerate the binary sequence b(k)={{circumflex over (b)}₀, {circumflexover (b)}₁, {circumflex over (b)}₂, . . . , {circumflex over (b)}_(k−b),{circumflex over (b)}_(k−1), {circumflex over (b)}_(k), {circumflex over(b)}_(k+1), . . . } 260 in response to the complex digital decimatedsignal z_(k,ĵ) 240 by using the frequency offset estimation {circumflexover (Ω)}_(Δ) 220, a first bit {circumflex over (b)}_(k−1) 270 and asecond bit {circumflex over (b)}_(k−2) 275, wherein the first bit{circumflex over (b)}_(k−1) 270 and the second bit {circumflex over(b)}_(k−2) 275 are generated by passing the binary sequenceb(k)={{circumflex over (b)}₀, {circumflex over (b)}₁, {circumflex over(b)}₂, . . . , {circumflex over (b)}_(k−2), {circumflex over (b)}_(k−1),{circumflex over (b)}_(k), {circumflex over (b)}_(k+1), . . . } througha T delay circuit 261 and two T delay circuit (261+262) respectively.

Finally, in FIG. 2, a frame synchronization circuit 280 receives thebinary sequence 260 b(k)={{circumflex over (b)}_(o), {circumflex over(b)}₁, {circumflex over (b)}₂, . . . , {circumflex over (b)}_(k−2),{circumflex over (b)}_(k−1), {circumflex over (b)}_(k), {circumflex over(b)}_(k+1), . . . } and uses the preamble data t(k) defined in the GFSKsystem, to obtain the starting bit 290 of the binary sequence 260b(k)={{circumflex over (b)}₀, {circumflex over (b)}₁, {circumflex over(b)}₂, . . . , {circumflex over (b)}_(k−2), {circumflex over (b)}_(k−1),{circumflex over (b)}_(k), {circumflex over (b)}_(k+1), . . . }.

In one exemplar aspect of the present invention, the carriersynchronization circuit 210 generates the frequency offset estimation{circumflex over (Ω)}_(Δ) 220 by performing the following steps shown inFIG. 3. Fist in step 300, the carrier synchronization circuit 210 storesthe first L complex baseband signal z_(k,j) 200, where L is apredetermined number, then step 310 the carrier synchronization circuit210 performs one-bit differential detection over the L symbols in thecomplex baseband signal z_(k,j) to obtain a parameter c_(k,j), step 320performs summation operation of the parameter c_(k,j), with respect tothe L symbols, and performs summation operation with respect to Nsampling points to obtain a complex value V. Step 330 estimates a biasΩ_(d) based on the preamble data t(k) defined in the GFSK system, it isnoted that step 330 can be executed before step 310 to get the biasΩ_(d) in advance. Step 340 calculates the frequency offset estimation{circumflex over (Ω)}_(Δ) 220 based on the bias Ω_(d) and the angle ofthe complex value V.

In one preferred embodiment of the present invention, the carriersynchronization circuit 210 stores the complex baseband signal z_(k,j)200 in step 300. In step 310, calculates c_(k,j)=z_(k+1,j)z_(k,j) ^(*),wherein the z_(k,j)* is the conjugate complex number of z_(k,j). In step320, obtain V by summation of c_(k,j) over L and N, wherein L is apredetermined number, step 330 estimate the bias from the preamble datat(k). {circumflex over (Ω)}_(d) is determined by N₁, N₂, N₃, N₄ and h,N₁ representing number of occurrences of b_(k)=b_(k+1)=1, N₂representing number of occurrences of b_(k)=b_(k+1)=−1, N₃ representingnumber of occurrences of b_(k)=1, b_(k+1)=−1, N₄ representing number ofoccurrences of b_(k)=−1, b_(k+1)=1, b_(k) is a value of a kth symbol ofthe preamble data t(k), h is the modulation index of the GFSKcommunication system.

N₁, N₂, N₃, N₄ can be obtained as followed, please refer to FIG. 4,store a first L symbols of the preamble data t(k) in a first memory X ofL words 400 (L is a predetermined number), provide a second memory N 410having a first word, a second word, a third word and a fourth word, eachword with an initial value of 0 and having an address comprise of baseaddress and an offset address, then repeating the following steps forL-1 times: (1) obtaining s by retrieving last 2 bits of a data in memoryX and do “and” operation with “11” (shown in 420); (2) adding 1 to thecontent of one of the first word, the second word, the third word andthe fourth word which its offset address equaling to s (shown in 420);(3) shifting the data in memory X right by one bit (as in 440). Afterthe repeating steps are done, obtain N₁, N₂, N₃, N₄ by retrievingcontents of the first word, the second word, the third word and thefourth word.

Finally, step 340 obtain the frequency offset estimation {circumflexover (Ω)}_(Δ) 220 by removing {circumflex over (Ω)}_(d) from arg(V) andnormalized by T, wherein arg(V) is an angle of V, T is a symbol time.

Therefore, this invention estimates and removes the bias embedded in thepreamble data t(k). And this invention can be applied to all ofcommunication systems as long as the preamble introduces a bias.

In one aspect of the present invention, wherein the clocksynchronization circuit 230 generates a complex digital decimated signalz_(k,ĵ) 240 by estimating a sampling point ĵ and performing thefollowing steps. Please refer to FIG. 5. Step 510 corrects c_(k,j) basedon {circumflex over (Ω)}_(Δ)T to obtain c^(f) _(k,j), then in Step 520obtains a first value u_(j) by summation of |Im(c^(f) _(k,j))| over L,wherein L is a predetermined number. In step 530, chooses the symboltiming point ĵ with the largest u_(j) by

${\hat{j} = {\arg\left\{ {\max\limits_{j}\; u_{j}} \right\}}};$step 540 outputs the complex digital decimated signal z_(k,ĵ) 240.

In the past, the symbol timing is obtained by calculating c^(f)_(i,j)=c_(k,j)·exp(−j·{circumflex over (Ω)}_(Δ)T) at first, whereinc_(k,j)=z_(k+1,j)z_(k,j) ^(*), z_(k,j)* is the conjugate complex numberof z_(k,j). Then obtains u_(j) by summation of |Im(c^(f) _(k,j))| overL, L is a predetermined number. This present invention simplifies theoperation by obtaining a first value A and a second value B by A=Ccos({circumflex over (Ω)}_(Δ)T), B=C sin({circumflex over (Ω)}_(Δ)T),wherein C is a constant, the obtain the value u_(j) by summation of|Im(c_(k,j))*A+Re(c_(k,j))*B| over L.

Please refer to FIG. 6, in one aspect of the present invention, thedemodulation circuit 250 demodulates the complex digital decimatedsignal z_(k,ĵ) to generate the binary sequence b(k)={{circumflex over(b)}₀, {circumflex over (b)}₁, {circumflex over (b)}₂, . . . ,{circumflex over (b)}_(k−2), {circumflex over (b)}_(k−1), {circumflexover (b)}_(k), {circumflex over (b)}_(k+1), . . . }, and performs thefollowing steps: Step 600 obtains a first angle θ using a first bit{circumflex over (b)}_(k−1) and the frequency offset estimation{circumflex over (Ω)}_(Δ) 220. Step 610 obtains a second angle Λ using asecond bit {circumflex over (b)}_(k−2) and the frequency offsetestimation {circumflex over (Ω)}_(Δ) 220; Perform one-bit differentialdetection with respect to the first bit {circumflex over (b)}_(k−1) toobtain a first value, and perform two-bit differential detection withrespect to the second bit {circumflex over (b)}_(k−2) to obtain a secondvalue in step 620, and obtain a complex sequence using the first value,the first angle θ, the second value, and the second angle Λ. Finally,step 630 obtains a binary sequence b(k)={{circumflex over (b)}₀,{circumflex over (b)}₁, {circumflex over (b)}₂, . . . , {circumflex over(b)}_(k−2), {circumflex over (b)}_(b−1), {circumflex over (b)}_(k),{circumflex over (b)}_(k+1), . . . } using an imaginary part of thecomplex sequence.

In one preferred embodiment of the invention. Obtaining an estimatedbinary sequence b(k)={{circumflex over (b)}₀, {circumflex over (b)}₁,{circumflex over (b)}₂, . . . , {circumflex over (b)}_(k−2), {circumflexover (b)}_(k−1), {circumflex over (b)}_(k), {circumflex over (b)}_(k+1),. . . } can be implemented by the following equations: obtain a firstangle θ by Θ=C₂{circumflex over (Ω)}_(Δ)T−h_(k−1)·πhδ; obtain a secondangle Λ by

$\Lambda = {{C_{3}\Omega_{\Delta}^{\hat{}}T} + \left\{ \begin{matrix}{{{- \pi}\; h} + {2{\delta\pi}\; h}} & {{b_{k - 2} = 0},{b_{k - 1} = 1}} \\{{\pi\; h} - {2{\delta\pi}\; h}} & {{b_{k - 2} = 1},{b_{k - 1} = 0},} \\{{- \pi}\;{h \cdot b_{k - 1}}} & {otherwise}\end{matrix} \right.}$wherein the parameter h is the modulation index defined in GFSK system,

${\delta = \frac{\int_{- T}^{0}{{p(t)}\ {\mathbb{d}t}}}{\int_{- \infty}^{\infty}{{p(t)}\ {\mathbb{d}t}}}},$which is derived from the a Gaussian function p(t) defined in the GFSKcommunication system; then obtain the estimated binary sequenceb(k)={{circumflex over (b)}₀, {circumflex over (b)}₁, {circumflex over(b)}₂, . . . , {circumflex over (b)}_(k−2), {circumflex over (b)}_(k−1),{circumflex over (b)}_(k), {circumflex over (b)}_(k+1), . . . } bywherein C₁, C₂, C₃ are constants.

In another aspect of the present invention, the frame synchronizationcircuit 280 obtains a starting bit 290 of the binary sequenceb(k)={{circumflex over (b)}₀, {circumflex over (b)}₁, {circumflex over(b)}₂, . . . , {circumflex over (b)}_(k−2), {circumflex over (b)}_(k−1),{circumflex over (b)}_(k), {circumflex over (b)}_(k+1), . . . } 260 byperforming the following steps shown in FIG. 7. It should be noted thatthe binary sequence b(k)={{circumflex over (b)}₀, {circumflex over(b)}₁, {circumflex over (b)}₂, . . . , {circumflex over (b)}_(k−2),{circumflex over (b)}_(k−1), {circumflex over (b)}_(k), {circumflex over(b)}_(k+1), . . . } 260 used herein is limited to unipolar binarysequence having values of {1, 0} in order to perform the XOR operationafterwards. In response to the binary sequence b(k)={{circumflex over(b)}₀, {circumflex over (b)}₁, {circumflex over (b)}₂, . . . ,{circumflex over (b)}_(k−2), {circumflex over (b)}_(k−1), {circumflexover (b)}_(k), {circumflex over (b)}_(k+1), . . . } 260 and the preambledata t(k) in the complex digital signal 200, step 700 obtains a value byperforming XOR-operation to the binary sequence b(k)={{circumflex over(b)}₀, {circumflex over (b)}₁, {circumflex over (b)}₂, . . . ,{circumflex over (b)}_(k−2), {circumflex over (b)}_(k−1), {circumflexover (b)}_(k), {circumflex over (b)}_(k+1), . . . } 260 and the preambledata t(k). Step 710 obtains a series of coefficients by generating a sumof the value, then choose a minimum value of the series of coefficientsto find out the starting bit 290 of the binary sequenceb(k)={{circumflex over (b)}₀, {circumflex over (b)}₁, {circumflex over(b)}₂, . . . , {circumflex over (b)}_(k−2), {circumflex over (b)}_(k−1),{circumflex over (b)}_(k), {circumflex over (b)}_(k+1), . . . } 260.FIG. 7 can be further implemented by obtaining a series of coefficientscoff(k) by summation of b(k+n)(xor)t(k) over M, wherein M is number ofbits, wherein (XOR) is an exclusive-or operation; and choose thestarting bit 290 of the binary sequence b(k)={{circumflex over (b)}₀,{circumflex over (b)}₁, {circumflex over (b)}₂, . . . , {circumflex over(b)}_(k−2), {circumflex over (b)}_(k−1), {circumflex over(b)}_(k),{circumflex over (b)}_(k+1), . . . } 260 by choosing theminimum of the series of coefficient by min(coff(k)).

In this example, because the present invention uses (XOR) operationinstead of multiplication, thus the invention uses minimum instead ofmaximum operation in finding the starting bit 290 of the binary sequenceb(k)={{circumflex over (b)}₀, {circumflex over (b)}₁, {circumflex over(b)}₂, . . . , {circumflex over (b)}_(k−2), {circumflex over (b)}_(k−1),{circumflex over (b)}_(k), {circumflex over (b)}_(k+1), . . . } 260.This method can be applied to all the systems as long as the bit streaminformation is in the form of unipolar binary (0, 1) sequence.

In the foregoing specification the invention has been described withreference to specific exemplar aspects thereof. It will, however, beevident that various modification and changes may be made to theretowithout departing from the broader spirit and scope of the invention.The specification and drawings are, accordingly, to be regarded in anillustrative rather than restrictive sense.

1. A baseband circuit for estimating a binary sequence b(k)={{circumflexover (b)}₀, {circumflex over (b)}₁, {circumflex over (b)}₂, . . . ,{circumflex over (b)}_(k−2), {circumflex over (b)}_(k−1), {circumflexover (b)}_(k), {circumflex over (b)}_(k+1), . . . } in a GFSK (GaussianFrequency Shift Keying) communication system, comprising: a carriersynchronization circuit, responsive to a complex baseband signal z_(k,j)and a preamble data t(k), for generating a frequency offset estimation{circumflex over (Ω)}_(Δ), wherein z_(k,j) representing a j-th sample ofa k-th symbol of a received data, 0≦j≦N−1, N representing samples persymbol; a clock synchronization circuit, responsive to the complexbaseband signal z_(k,j) and the frequency offset estimation {circumflexover (Ω)}₆₆, for generating a complex digital decimated signal z_(k,ĵ)by estimating a sampling point ĵ; a demodulation circuit with decisionfeedback, responsive to the complex digital decimated signal z_(k,ĵ),the frequency offset estimation {circumflex over (Ω)}_(Δ), a first bit{circumflex over (b)}_(k−1) and a second bit {circumflex over(b)}_(k−2), for demodulating the complex digital decimated signalz_(k,ĵ) to generate the binary sequence b(k)={{circumflex over (b)}₀,{circumflex over (b)}₁, {circumflex over (b)}₂, . . . , {circumflex over(b)}_(k−2), {circumflex over (b)}_(k−1), {circumflex over (b)}_(k),{circumflex over (b)}_(k+1), . . . }; and a frame synchronizationcircuit, responsive to the binary sequence b(k)={{circumflex over (b)}₀,{circumflex over (b)}₁, {circumflex over (b)}₂, . . . , {circumflex over(b)}_(k−2), {circumflex over (b)}_(k−1), {circumflex over (b)}_(k),{circumflex over (b)}_(k+1), . . . } and the preamble data t(k), forobtaining a starting bit {circumflex over (b)}_({circumflex over (k)})of the binary sequence b(k)={{circumflex over (b)}₀, {circumflex over(b)}₁, {circumflex over (b)}₂, . . . , {circumflex over (b)}_(k−2),{circumflex over (b)}_(k−1), {circumflex over (b)}_(k), {circumflex over(b)}_(k+1), . . . }.
 2. A baseband circuit for estimating a binarysequence b(k)={{circumflex over (b)}₀, {circumflex over (b)}₁,{circumflex over (b)}₂, . . . , {circumflex over (b)}_(k−2), {circumflexover (b)}_(k−1), {circumflex over (b)}_(k), {circumflex over (b)}_(k+1),. . . } in Gaussian Frequency Shift communication system, comprising: acarrier synchronization circuit for generating a frequency offsetestimation {circumflex over (Ω)}_(Δ) by: storing a complex basebandsignal z_(k,j), wherein z_(k,j) representing a j-th sample of a k-thsymbol of a received data, 0≦j≦N−1, N representing samples per symbol;performing one-bit differential detection over L symbols in the complexbaseband signal z_(k,j) to obtain a parameter c_(k,j), wherein L is apredetermined number; performing summation operation of the c_(k,j) withrespect to the first L symbols, and performing summation operation withrespect to N sampling points, to obtain a complex value V; estimating abias {circumflex over (Ω)}_(d) based on a preamble data t(k);calculating the frequency offset estimation {circumflex over (Ω)}_(Δ)based on the bias {circumflex over (Ω)}_(d) and an angle of the complexvalue V; a clock synchronization circuit for generating a complexdigital decimated signal z_(k,ĵ) by estimating a sampling point ĵ usingthe frequency offset estimation {circumflex over (Ω)}_(Δ) and thecomplex baseband signal z_(k,j); and a demodulation circuit, responsiveto the complex digital decimated signal z_(k,ĵ) and the frequency offsetestimation {circumflex over (Ω)}_(Δ) for demodulating the complexdigital decimated signal z_(k,ĵ) to generate the binary sequenceb(k)={{circumflex over (b)}₀, {circumflex over (b)}₁, {circumflex over(b)}₂, . . . , {circumflex over (b)}_(k−2), {circumflex over (b)}_(k−1),{circumflex over (b)}_(k), {circumflex over (b)}_(k+1), . . . }.
 3. Thecircuit of claim 2, wherein {circumflex over (Ω)}_(d) is furtherdetermined by N₁, N₂, N₃, N₄ and h, wherein N₁ representing number ofoccurrences of b_(k)=b_(b+1)=1, N₂ representing number of occurences ofb_(k=)b_(k+1)=−1, N₃ representing number of occurrences of b_(k)=1,b_(k+1)=−1, N₄ representing number of occurrences of b_(k)=−1, b_(k) isa value of a kth symbol of the preamble data t(k), h is a modulationindex of the GFSK communication system.
 4. The circuit of claim 3,wherein N₁, N₂, N₃, N₄ are determined by: storing a first L symbols ofthe preamble data t(k) in a first memory X of L words, L is apredetermined number; providing a second memory N having a first word, asecond word, a third word and a fourth word, each word having a baseaddress and an offset address with an initial value of 0; repeating thefollowing steps for L-1 times: (40.1) obtaining s by retrieving last 2bits of a data in memory X and do “and” operation with “11”; (40.2)adding 1 to a content of one of the first word, the second word, thethird word and the fourth word which has offset address equaling to s;(40.3) shifting the data in memory X right by one bit; obtaining N₁, N₂,N₃, N₄ by retrieving contents of the first word, the second word, thethird word and the fourth word.
 5. The circuit of claim 2, wherein{circumflex over (Ω)}_(Δ) is further determined by removing {circumflexover (Ω)}_(d) from arg(V) and normalized by T, wherein arg(V) is anangle of V, T is a symbol time.
 6. The circuit of claim 2, wherein thedemodulation circuit is further responsive to a first bit {circumflexover (b)}_(k−1) and a second bit {circumflex over (b)}_(k−2).
 7. Abaseband circuit for estimating a binary sequence b(k)={{circumflex over(b)}₀, {circumflex over (b)}₁, {circumflex over (b)}₂, . . . ,{circumflex over (b)}_(k−2), {circumflex over (b)}_(k−1), {circumflexover (b)}_(k), {circumflex over (b)}_(k+1), . . . } in a GFSK (GaussianFrequency Shift Keying) communication system, comprising: a carriersynchronization circuit for generating a frequency offset estimation{circumflex over (Ω)}_(Δ) and performing one-bit differential detectionof a first L symbols in a complex baseband signal z_(k,j) to obtain aparameter c_(k,j), wherein L is a predetermined number, and z_(k,j)representing a j-th sample of a k-th symbol of a received data, 0≦j≦N−1,and N represents samples per symbol; a clock synchronization circuit forgenerating a complex digital decimated signal z_(k,ĵ) by: correctingc_(k,j) based on {circumflex over (Ω)}_(Δ)T to obtain c^(f) _(k,j);wherein T is a symbol obtaining a first value u_(j) by summation of|Im(c^(f) _(k,j))| over L; and choosing the symbol timing ĵ with thelargest u_(j) by${\hat{j} = {\arg\left\{ {\max\limits_{j}\; u_{j}} \right\}}};$ ademodulation circuit, responsive to the complex digital decimated signalz_(k,ĵ) and the frequency offset estimation {circumflex over (Ω)}_(Δ)for demodulating the complex digital decimated signal z_(k,ĵ) togenerate the binary sequence b(k)={{circumflex over (b)}₀, {circumflexover (b)}₁, {circumflex over (b)}₂, . . . , {circumflex over (b)}_(k−2),{circumflex over (b)}_(k−1), {circumflex over (b)}_(k), {circumflex over(b)}_(k+1), . . . }.
 8. The circuit of claim 7, wherein the first valueu_(j) is further determined by: obtaining a first value A and a secondvalue B by A=C cos({circumflex over (Ω)}_(Δ)T), B=C sin({circumflex over(Ω)}_(Δ)T), wherein C is a constant; obtaining a third value u_(j) bysummation of |Im(c_(k,j))*A+Re(c_(k,j))(B| over L, wherein L is apredetermined number.
 9. The circuit of claim 7, wherein thedemodulation circuit is further responsive to a first bit {circumflexover (b)}_(k−1) and a second bit {circumflex over (b)}_(k−2).
 10. Abaseband circuit for estimating a binary sequence b(k){{circumflex over(b)}₀, {circumflex over (b)}₁, {circumflex over (b)}₂, . . . ,{circumflex over (b)}_(k−2), {circumflex over (b)}_(k−1), {circumflexover (b)}_(k), {circumflex over (b)}_(k+1), . . . } in a GFSK (GaussianFrequency Shift Keying) communication system, comprising: a carriersynchronization circuit, responsive to a complex baseband signal z_(k,j)and a preamble data t(k), for generating a frequency offset estimation{circumflex over (Ω)}_(Δ), wherein z_(k,j) representing a j-th sample ofa k-th symbol of a received data, 0≦j≦N−1, N representing samples persymbol; a clock synchronization circuit, responsive to the complexbaseband signal z_(k,j) and the estimated frequency offset {circumflexover (Ω)}_(Δ), for generating a complex digital decimated signal z_(k,ĵ)by estimating a sampling point ĵ; and a demodulation circuit fordemodulating the complex digital decimated signal z_(k,ĵ) to generatethe binary sequence b(k)={{circumflex over (b)}₀, {circumflex over(b)}₁, {circumflex over (b)}₂, . . . , {circumflex over (b)}_(k−2),{circumflex over (b)}_(k−1), {circumflex over (b)}_(k), {circumflex over(b)}_(k+1), . . . } by: obtaining a first angle θ using a first bit{circumflex over (b)}_(k−1) and the frequency offset estimation{circumflex over (Ω)}_(Δ); obtaining a second angle Λ using a second bit{circumflex over (b)}_(k−2) and the frequency offset estimation{circumflex over (Ω)}_(Δ); performing one-bit differential detectionwith respect to the first bit {circumflex over (b)}_(k−1) to obtain afirst value, and performing two-bit differential detection with respectto the second bit {circumflex over (b)}_(k−2) to obtain a second value,obtaining a complex sequence using the first value, the first angle Θ,the second value, and the second angle Λ; obtaining the binary sequenceb(k){{circumflex over (b)}₀, {circumflex over (b)}₁, {circumflex over(b)}₂, . . . , {circumflex over (b)}_(k−2), {circumflex over (b)}_(k−1),{circumflex over (b)}_(k), {circumflex over (b)}_(k+1), . . . } using animaginary part of the complex sequence.
 11. The circuit of claim 10,wherein Θ is obtained by Θ=C₂{circumflex over (Ω)}_(Δ)T−b_(k−1)·πhδ, his a demodulation index in the GFSK communication system,${\delta = \frac{\int_{- T}^{0}{{p(t)}{\mathbb{d}t}}}{\int_{- \infty}^{\infty}{{p(t)}{\mathbb{d}t}}}},$is a Gaussian function, T is a symbol time; Λ is obtained$\Lambda = {{C_{3}\hat{\Omega_{\Delta}}T} + \left\{ \begin{matrix}{{{- \pi}\; h} + {2{\delta\pi}\; h}} & {{b_{k - 2} = 0},{b_{k - 1} = 1}} \\{{\pi\; h} - {2{\delta\pi}\; h}} & {{b_{k - 2} = 1},{{b_{k - 1} = 0};}} \\{{- \pi}\;{h \cdot b_{k - 1}}} & {otherwise}\end{matrix} \right.}$ b(k)={{circumflex over (b)}₀, {circumflex over(b)}₁, {circumflex over (b)}₂, . . . , {circumflex over (b)}_(k−2),{circumflex over (b)}_(k−1), {circumflex over (b)}_(k), {circumflex over(b)}_(k+1), . . . } is obtained by{circumflex over (b)} _(k) =sgn[Im(z _(k+1,ĵ) ·z _(k,ĵ) ^(*) ·e ^(jΘ) +C₁ ·z _(k+1,ĵ) ·z _(k−1,ĵ) ·e ^(jΛ))], wherein C₁, C₂, C₃ are constants,z_(k,ĵ) * is a conjugate complex number of z_(k,ĵ), and “sgn” representstaking sign of a quantity by “0” for a negative sign and “1” for apositive sign and zero.
 12. A baseband circuit for estimating a unipolarbinary sequence b(k)={{circumflex over (b)}₀, {circumflex over (b)}₁,{circumflex over (b)}₂, . . . , {circumflex over (b)}_(k−2), {circumflexover (b)}_(k−1), {circumflex over (b)}_(k), {circumflex over (b)}_(k+1),. . . } having values of {0, 1} in a GFSK (Gaussian Frequency ShiftKeying) communication system, comprising: a demodulation circuit forgenerating a binary sequence b(k)={{circumflex over (b)}₀, {circumflexover (b)}₁, {circumflex over (b)}₂, . . . , {circumflex over (b)}_(b−2),{circumflex over (b)}_(k−1), {circumflex over (b)}_(k), {circumflex over(b)}_(k+1), . . . }; and a frame synchronization circuit for obtaining astarting bit {circumflex over (b)}_({circumflex over (k)}) of the binarysequence b(k)={{circumflex over (b)}₀, {circumflex over (b)}₁,{circumflex over (b)}₂, . . . , {circumflex over (b)}_(k−2), {circumflexover (b)}_(k−1), {circumflex over (b)}_(k), {circumflex over (b)}_(k+1),. . . } by: responsive to the binary sequence b(k)={{circumflex over(b)}₀, {circumflex over (b)}₁, {circumflex over (b)}₂, . . . ,{circumflex over (b)}_(k−2), {circumflex over (b)}_(k−1), {circumflexover (b)}_(k), {circumflex over (b)}_(k+1), . . . } and a preample datat(k), obtaining a value by performing XOR-operation to the binarysequence b(k)={{circumflex over (b)}₀, {circumflex over (b)}₁,{circumflex over (b)}₂, . . . , {circumflex over (b)}_(k−2), {circumflexover (b)}_(k−1), {circumflex over (b)}_(k), {circumflex over (b)}_(k+1),. . . } and the preamble data t(k); obtaining a series of coefficientsby generating a sum of the values; and choosing the starting bit{circumflex over (b)}_({circumflex over (k)}) of the binary sequenceb(k)={{circumflex over (b)}₀, {circumflex over (b)}₁, {circumflex over(b)}₂, . . . , {circumflex over (b)}_(k−2), {circumflex over (b)}_(k−1),{circumflex over (b)}_(k), {circumflex over (b)}_(k+1), . . . } bychoosing a minimum value of the series of coefficients.
 13. A basebandcircuit for estimating a unipolar binary sequence b(k)={{circumflex over(b)}₀, {circumflex over (b)}₁, {circumflex over (b)}₂, . . . ,{circumflex over (b)}_(l−2), {circumflex over (b)}_(l−1), {circumflexover (b)}_(k), {circumflex over (b)}_(k+1), . . . } having values of {0,1} in a GFSK (Gaussian Frequency Shift Keying) communication system,comprising: a carrier synchronization circuit for generating a frequencyoffset estimation {circumflex over (Ω)}_(Δ) by: storing a complexbaseband signal z_(k,j), wherein z_(k,j) representing a j-th sample of ak-th symbol of a received data, 0≦j≦N−1, N representing samples persymbol; calculating c_(k,j)=z_(k+1,j)z_(k,j) ^(*), wherein the z_(k,j)*is the conjugate complex number of z_(k,j); obtaining V by summation ofc_(k,j) over L and N, wherein L is a predetermined number; and obtainingthe frequency offset estimation {circumflex over (Ω)}_(Δ) by removing{circumflex over (Ω)}_(d) from arg(V) and normalized by T, whereinarg(V) is an angle of V, T is a symbol time, {circumflex over (Ω)}_(d)is determined by N₁, N₂, N₃, N₄ and h, N₁ representing number ofoccurrences of b_(k)=b_(k+1)=1, N₂ representing number of occurrences ofb_(k)=b_(k+1)=−1, N₃ representing number of occurrences of b_(k)=1,b_(k+1)=−1, N₄ representing number of occurrences of b_(k)−1, b_(k+1)=1,b_(k) is a value of a kth symbol of the preamble data t(k), h is amodulation index of the GFSK communication system; a clocksynchronization circuit for generating a complex digital decimatedsignal z_(k,ĵ) by: obtaining a first value A and a second value B by A=Ccos({circumflex over (Ω)}_(Δ)T), B=C sin({circumflex over (Ω)}_(Δ)T);wherein C is a constant; obtaining a third value u_(j) by summation of|Im(c_(k,j))*A+Re(c_(k,j))*B| over L; and choosing a symbol timing ĵwith the largest u_(j) by${\hat{j} = {\arg\left\{ {\max\limits_{j}\; u_{j}} \right\}}};$ ademodulation circuit for demodulating the complex digital decimatedsignal z_(k,ĵ) to generate the binary sequence b(k)={{circumflex over(b)}₀, {circumflex over (b)}₁, {circumflex over (b)}₂, . . . ,{circumflex over (b)}_(k−2), {circumflex over (b)}_(k−1), {circumflexover (b)}_(k), {circumflex over (b)}_(k+1), . . . } by: obtaining afirst angle θ by Θ=C₂{circumflex over (Ω)}_(Δ)T−b_(k−1)·πhδ, wherein his a demodulation index in the GFSK communication system,${\delta = \frac{\int_{- T}^{0}{{p(t)}{\mathbb{d}t}}}{\int_{- \infty}^{\infty}{{p(t)}{\mathbb{d}t}}}},$is a Gaussian function; obtaining a second angle Λ by$\Lambda = {{C_{3}\hat{\Omega_{\Delta}}T} + \left\{ \begin{matrix}{{{- \pi}\; h} + {2{\delta\pi}\; h}} & {{b_{k - 2} = 0},{b_{k - 1} = 1}} \\{{\pi\; h} - {2{\delta\pi}\; h}} & {{b_{k - 2} = 1},{{b_{k - 1} = 0};\mspace{14mu}{and}}} \\{{- \pi}\;{h \cdot b_{k - 1}}} & {otherwise}\end{matrix} \right.}$ obtaining b(k)={{circumflex over (b)}₀,{circumflex over (b)}₁, {circumflex over (b)}₂, . . . , {circumflex over(b)}_(k−2), {circumflex over (b)}_(k−1), {circumflex over (b)}_(k),{circumflex over (b)}_(k+1), . . . } by{circumflex over (b)} _(k) =sgn[Im(z _(k+1,ĵ) ·z _(k,ĵ) ^(*) ·e ^(jΘ) +C₁ ·z _(k+1,ĵ)  z _(k−1,ĵ) ^(*) ·e ^(jΛ))], wherein C₁, C₂, C₃ areconstants, z_(k,ĵ) * is the conjugate complex number of z_(k,ĵ), and“sgn” represents taking sign of a quantity by “0” for a negative signand “1” for a positive sign and zero; a frame synchronization circuitfor obtaining a starting bit {circumflex over(b)}_({circumflex over (k)}) of the binary sequence b(k)={{circumflexover (b)}₀, {circumflex over (b)}₁, {circumflex over (b)}₂, . . . ,{circumflex over (b)}_(k−2), {circumflex over (b)}_(k−1), {circumflexover (b)}_(k), {circumflex over (b)}_(k+1), . . . } by: responsive tothe binary sequence b(k)={{circumflex over (b)}₀, {circumflex over(b)}₁, {circumflex over (b)}₂, . . . , {circumflex over (b)}_(k−2),{circumflex over (b)}_(k−1), {circumflex over (b)}_(k), {circumflex over(b)}_(k+1), . . . } and the preamble data t(k), obtaining a series ofcoefficient coff(k) by summation of b(l+n)(xor)t(k) over M, wherein M isnumber of bits; and choosing the starting bit {circumflex over(b)}_({circumflex over (k)}) of the binary sequence b(k)={{circumflexover (b)}₀, {circumflex over (b)}₁, {circumflex over (b)}₂, . . . ,{circumflex over (b)}_(k−2), {circumflex over (b)}_(k−1), {circumflexover (b)}_(k), {circumflex over (b)}_(k+1), . . . } by min(coff(k)). 14.A method for estimating a binary sequence b(k)={{circumflex over (b)}₀,{circumflex over (b)}₁, {circumflex over (b)}₂, . . . , {circumflex over(b)}_(k−2), {circumflex over (b)}_(k−1), {circumflex over (b)}_(k),{circumflex over (b)}_(k+1), . . . } in a GFSK (Gaussian Frequency ShiftKeying) communication system, comprising the following steps: (14.1)responsive to a complex baseband signal z_(k,j) and a preamble datat(k), generating a frequency offset estimation {circumflex over(Ω)}_(Δ), wherein z_(k,j) representing a j-th sample of a k-th symbol ofa received data, 0≦j≦N−1, N representing samples per symbol; (14.2)responsive to the complex baseband signal z_(k,j) and the frequencyoffset estimation {circumflex over (Ω)}_(Δ), generating a complexdigital decimated signal z_(k,ĵ) by estimating a sampling point ĵ;(14.3) responsive to the complex digital decimated signal z_(k,ĵ), thefrequency offset estimation {circumflex over (Ω)}_(Δ), a first bit{circumflex over (b)}_(k−1) and a second bit {circumflex over(b)}_(k−2), demodulating the complex digital decimated signal z_(k,ĵ) togenerate the binary sequence b(k)={{circumflex over (b)}₀, {circumflexover (b)}₁, {circumflex over (b)}₂, . . . , {circumflex over (b)}_(k−2),{circumflex over (b)}_(k−1), {circumflex over (b)}_(k), {circumflex over(b)}_(k+1), . . . }; and (14.4) frame synchronization circuit,responsive to the binary sequence b(k)={{circumflex over (b)}₀,{circumflex over (b)}₁, {circumflex over (b)}₂, . . . , {circumflex over(b)}_(k−2), {circumflex over (b)}_(k−1), {circumflex over (b)}_(k),{circumflex over (b)}_(k+1), . . . } and the preamble data t(k), forobtaining a starting bit {circumflex over (b)}_({circumflex over (k)})of the binary sequence b(k)={{circumflex over (b)}₀, {circumflex over(b)}₁, {circumflex over (b)}₂, . . . , {circumflex over (b)}_(k−2),{circumflex over (b)}_(k−1), {circumflex over (b)}_(k), {circumflex over(b)}_(k+1), . . . }.
 15. A method for estimating a binary sequenceb(k)={{circumflex over (b)}₀, {circumflex over (b)}₁, {circumflex over(b)}₂, . . . , {circumflex over (b)}_(k−2), {circumflex over (b)}_(k−1),{circumflex over (b)}_(k), {circumflex over (b)}_(k+1), . . . } in aGFSK (Gaussian Frequency Shift Keying) communication syste, comprisingthe following steps: (15.1) obtaining a frequency offset estimation{circumflex over (Ω)}_(Δ) by performing the following steps: a. storinga first L complex baseband signal z_(k,j), wherein z_(k,j) representinga j-th sample of a k-th symbol of a received data, 0≦j≦N−1, Nrepresenting samples per symbol; and L is a predetermined number; b.performing one-bit differential detection of a first L symbols in thecomplex baseband signal z_(k,j) to obtain a parameter c_(k,j); c.performing summation operation of the c_(k,j) with respect to the firstL symbols, and performing summation operation with respect to N samplingpoints, to obtain a complex value V; d. estimating a bias {circumflexover (Ω)}_(d) based on a preamble data t(k); and e. calculating thefrequency offset estimation {circumflex over (Ω)}_(Δ) based on the bias{circumflex over (Ω)}_(d) and an angle of the complex value V; (15.2)generating a complex digital decimated signal z_(k,ĵ) by estimating asampling point ĵ using the frequency offset estimation {circumflex over(Ω)}_(Δ) and the complex baseband signal z_(k,j); and (15.3) responsiveto the complex digital decimated signal z_(k,ĵ) and the frequency offsetestimation {circumflex over (Ω)}_(Δ), demodulating the complex digitaldecimated signal z_(k,ĵ) to generate the binary sequenceb(k)={{circumflex over (b)}₀, {circumflex over (b)}₁, {circumflex over(b)}₂, . . . , {circumflex over (b)}_(k−2), {circumflex over (b)}_(k−1),{circumflex over (b)}_(k), {circumflex over (b)}_(k+1), . . . }.
 16. Themethod of claim 15, wherein {circumflex over (Ω)}_(d) is furtherdetermined by N₁, N₂, N₃, N₄ and h, N₁ representing number ofoccurrences of b_(k)=b_(k+1)=1, N₂ representing number of occurrences ofb_(k)=b_(k+1)=−1, N₃ representing number of occurrences of b_(k)=1,b_(k+1)=−1, N₄ representing number of occurrences of b_(k)=−1, b_(k) isa value of a kth symbol of the preamble data t(k), h is a modulationindex of the GFSK communication system.
 17. The method of claim 16,wherein N₁, N₂, N₃, N₄ are determined by the following steps: (17.1)storing a first L symbols of the preamble data t(k) in a first memory Xof L words, wherein L is a predetermined number; (17.2) providing asecond memory N having a first word, a second word, a third word and afourth word, each word having a base address and an offset address withan initial value of 0; (17.3) repeating the following steps for L-1times: (170.1) obtaining s by retrieving last 2 bits of a data in memoryX and do “and” operation with “11”; (170.2) adding 1 to a content of oneof the first word, the second word, the third word and the fourth wordwhich has offset address equaling to s; (170.3) shifting the data inmemory X right by one bit; (17.4) obtaining N₁, N₂, N₃, N₄ by retrievingcontents of the first word, the second word, the third word and thefourth word.
 18. The method of claim 15, wherein {circumflex over(Ω)}_(Δ) is further determined by removing {circumflex over (Ω)}_(d)from arg(V) and normalized by T, arg(V) is an angle of V, T is a symboltime.
 19. The method of claim 15, wherein in the step (15.3), thegeneration of the binary sequence b(k)={{circumflex over (b)}₀,{circumflex over (b)}₁, {circumflex over (b)}₂, . . . , {circumflex over(b)}_(k−2), {circumflex over (b)}_(k−1), {circumflex over (b)}_(k),{circumflex over (b)}_(k+1), . . . } is further responsive to a firstbit {circumflex over (b)}_(k−1) and a second bit {circumflex over(b)}_(k−2).
 20. A method for estimating a binary sequenceb(k)={{circumflex over (b)}₀, {circumflex over (b)}₁, {circumflex over(b)}₂, . . . , {circumflex over (b)}_(k−2), {circumflex over (b)}_(k−1),{circumflex over (b)}_(k), {circumflex over (b)}_(k+1), . . . } in aGFSK (Gaussian Frequency Shift Keying) communication system, comprisingthe following steps: (20.1) generating a frequency offset estimation{circumflex over (Ω)}_(Δ) and performing one-bit differential detectionof a first L symbols in the complex baseband signal z_(k,j) to obtain aparameter c_(k,j), wherein z_(k,j) represents a j-th sample of a k-thsymbol of a received data, 0≦j≦N−1, N representing samples per symbol;and L is a predetermined number; (20.2) generating a complex digitaldecimated signal z_(k,ĵ) by: correcting c_(k,j) based on {circumflexover (Ω)}_(Δ)T to obtain c^(f) _(k,j), wherein T is a symbol time;obtaining a first value u_(j) by summation of |Im(c^(f) _(k,j))| over L;and choosing a symbol timing ĵ with the largest u_(j) by${\hat{j} = {\arg\left\{ {\max\limits_{j}\; u_{j}} \right\}}};$ (20.3)responsive to the complex digital decimated signal z_(k,ĵ) and thefrequency offset estimation {circumflex over (Ω)}_(Δ), demodulating thecomplex digital decimated signal z_(k,ĵ) to generate the binary sequenceb(k)={{circumflex over (b)}₀, {circumflex over (b)}₁, {circumflex over(b)}₂, . . . , {circumflex over (b)}_(k−2), {circumflex over (b)}_(k−1),{circumflex over (b)}_(k), {circumflex over (b)}_(k+1), . . . }.
 21. Themethod of claim 20, wherein the first value u_(j) is further determinedby: obtaining a first value A and a second value B by A=Ccos({circumflex over (Ω)}_(Δ)T), B=C sin({circumflex over (Ω)}_(Δ)T);wherein C is a constant; obtaining a third value u_(j) by summation of|Im(c_(k,j))*A+Re(c_(k,j))*B| over L, wherein L is a predeterminednumber.
 22. The method of claim 20, wherein the step (20.3) thegeneration of b(k)={{circumflex over (b)}₀, {circumflex over (b)}₁,{circumflex over (b)}₂, . . . , {circumflex over (b)}_(k−2), {circumflexover (b)}_(k−1), {circumflex over (b)}_(k), {circumflex over (b)}_(k+1),. . . } further responsive to a first bit {circumflex over (b)}_(k−1)and a second bit {circumflex over (b)}_(k−2).
 23. A method forestimating a binary sequence b(k)={{circumflex over (b)}₀, {circumflexover (b)}₁, {circumflex over (b)}₂, . . . , {circumflex over (b)}_(k−2),{circumflex over (b)}_(k−1), {circumflex over (b)}_(k), {circumflex over(b)}_(k+1), . . . } in a GFSK (Gaussian Frequency Shift Keying)communication system, comprising the following steps: (23.1) responsiveto a complex baseband signal z_(k,j) and a preamble data t(k),generating a frequency offset estimation {circumflex over (Ω)}_(Δ),wherein z_(k,j) representing a j-th sample of a k-th symbol of areceived data, 0≦j≦N−1, N representing samples per symbol; (23.2)responsive to the complex baseband signal z_(k,j) and the estimatedfrequency offset {circumflex over (Ω)}_(Δ), generating a complex digitaldecimated signal z_(k,ĵ) by estimating a sampling point ĵ; and (23.3)demodulating the complex digital decimated signal z_(k,ĵ) to generatethe binary sequence b(k)={{circumflex over (b)}₀, {circumflex over(b)}₁, {circumflex over (b)}₂, . . . , {circumflex over (b)}_(k−2),{circumflex over (b)}_(k−1), {circumflex over (b)}_(k), {circumflex over(b)}_(k+1), . . . } by: a. obtaining a first angle θ using a first bit{circumflex over (b)}_(k−1) and the frequency offset estimation{circumflex over (Ω)}_(Δ); b. obtaining a second angle Λ using a secondbit {circumflex over (b)}_(k−2) and the frequency offset estimation{circumflex over (Ω)}_(Δ); c. performing one-bit differential detectionwith respect to the first bit {circumflex over (b)}_(k−1) to obtain afirst value, and performing two-bit differential detection with respectto the second bit {circumflex over (b)}_(k−2) to obtain a second value,obtaining a complex sequence using the first value, the first angle θ,the second value, and the second angle Λ; d. obtaining the binarysequence b(k)={{circumflex over (b)}₀, {circumflex over (b)}₁,{circumflex over (b)}₂, . . . , {circumflex over (b)}_(k−2), {circumflexover (b)}_(k−1), {circumflex over (b)}_(k), {circumflex over (b)}_(k+1),. . . } using an imaginary part of the complex sequence.
 24. The methodof claim 23, wherein the step (23.3), Θ is further obtained byΘ=C₂{circumflex over (Ω)}_(Δ)T−b_(k−1)·πhδ, h is a demodulation index inthe GFSK communication system${\delta = \frac{\int_{- T}^{0}{{p(t)}{\mathbb{d}t}}}{\int_{- \infty}^{\infty}{{p(t)}{\mathbb{d}t}}}},$is a Gaussian function, T is a symbol time; Λ is obtained by$\Lambda = {{C_{3}\hat{\Omega_{\Delta}}T} + \left\{ \begin{matrix}{{{- \pi}\; h} + {2{\delta\pi}\; h}} & {{b_{k - 2} = 0},{b_{k - 1} = 1}} \\{{\pi\; h} - {2{\delta\pi}\; h}} & {{b_{k - 2} = 1},{{b_{k - 1} = 0};}} \\{{- \pi}\;{h \cdot b_{k - 1}}} & {otherwise}\end{matrix} \right.}$ b(k)={{circumflex over (b)}₀, {circumflex over(b)}₁, {circumflex over (b)}₂, . . . , {circumflex over (b)}_(k−2),{circumflex over (b)}_(k−1), {circumflex over (b)}_(k), {circumflex over(b)}_(k+1), . . . } is obtained by{circumflex over (b)} _(k) =sgn[Im(z _(k+1,ĵ) −z _(k,ĵ) ^(*) ·e ^(jΘ) +C₁ ·z _(k+1,ĵ) ·z _(k−1,ĵ) ^(*)  e ^(jΛ))], wherein C₁, C₂, C₃ areconstants, z_(k,ĵ) * is a conjugate complex number of z_(k,ĵ), and “sgn”represents taking sign of a quantity by “0” for a negative sign and “1”for a positive sign and zero.
 25. A method for estimating a unipolarbinary sequence b(k)={{circumflex over (b)}₀, {circumflex over (b)}₁,{circumflex over (b)}₂, . . . , {circumflex over (b)}_(k−2), {circumflexover (b)}_(k−1), {circumflex over (b)}_(k), {circumflex over (b)}_(k+1),. . . } having values of {0, 1} in a Gaussian Frequency Shift Keyingcommunication system, comprising the following steps: (25.1) generatinga binary sequence b(k)={{circumflex over (b)}₀, {circumflex over (b)}₁,{circumflex over (b)}₂, . . . , {circumflex over (b)}_(k−2), {circumflexover (b)}_(k−1), {circumflex over (b)}_(k), {circumflex over (b)}_(k+1),. . . }; and (25.2) obtaining a starting bit {circumflex over(b)}_({circumflex over (k)}) of the binary sequence b(k)={{circumflexover (b)}₀, {circumflex over (b)}₁, {circumflex over (b)}₂, . . . ,{circumflex over (b)}_(k−2), {circumflex over (b)}_(k−1), {circumflexover (b)}_(k), {circumflex over (b)}_(k+1), . . . } by: a. responsive tothe binary sequence b(k)={{circumflex over (b)}₀, {circumflex over(b)}₁, {circumflex over (b)}₂, . . . , {circumflex over (b)}_(k−2),{circumflex over (b)}_(k−1), {circumflex over (b)}_(k), {circumflex over(b)}_(k+1), . . . } and a preamble data t(k), obtaining a value byperforming XOR-operation to the binary sequence b(k)={{circumflex over(b)}₀, {circumflex over (b)}₁, {circumflex over (b)}₂, . . . ,{circumflex over (b)}_(k−2), {circumflex over (b)}_(k−1), {circumflexover (b)}_(k), {circumflex over (b)}_(k+1), . . . } and the preambledata t(k); b. obtaining a series of coefficient by generating a sum ofthe values; and choosing the starting bit {circumflex over(b)}_({circumflex over (k)}) of the binary sequence b(k)={{circumflexover (b)}₀, {circumflex over (b)}₁, {circumflex over (b)}₂, . . . ,{circumflex over (b)}_(k−2), {circumflex over (b)}_(k−1), {circumflexover (b)}_(k), {circumflex over (b)}_(k+1), . . . } by choosing aminimum value of the series of coefficient.
 26. A method for estimatinga unipolar binary sequence b(k)={{circumflex over (b)}₀, {circumflexover (b)}₁, {circumflex over (b)}₂, . . . , {circumflex over (b)}_(k−2),{circumflex over (b)}_(k−1), {circumflex over (b)}_(k), {circumflex over(b)}_(k+1), . . . } having values of {0, 1} in a Gaussian FrequencyShift Keying communication system, comprising the following steps:(26.1) generating a frequency offset estimation {circumflex over(Ω)}_(Δ) by: a. storing a complex baseband signal z_(k,j), whereinz_(k,j) representing a j-th sample of a k-th symbol of a received data,0≦j≦N−1, N representing samples per symbol; b. calculatingc_(k,j)=z_(k+1,j)z_(k,j) ^(*), wherein the z_(k,j)* is the conjugatecomplex number of z_(k,j); c. obtaining V by summation of c_(k,j) over Land N, wherein L is a predetermined number; d. obtaining the frequencyoffset estimation {circumflex over (Ω)}_(Δ) by removing {circumflex over(Ω)}_(d) from arg(V) and normalized by T, wherein arg(V) is an angle ofV, T is a symbol time, {circumflex over (Ω)}_(d) is determined by N₁,N₂, N₃, N₄ and h, N₁ representing number of occurrences ofb_(k)=b_(k+1)=1, N₂ representing number of occurrences ofb_(k)=b_(k+1)=−1, N₃ representing number of occurrences of b_(k)=1,b_(k+1)=−1, N₄ representing number of occurrences of b_(k)=−1,b_(k+1)=1, b_(k) is a value of a kth symbol of the preamble data t(k), his a modulation index of the GFSK communication system; (26.2)generating a complex digital decimated signal z_(k,ĵ) by: a. obtaining afirst value A and a second value B by A=C cos({circumflex over(Ω)}_(Δ)T), B=C sin({circumflex over (Ω)}_(Δ)T), wherein C is aconstant; b. obtaining a third value u_(j) by summation of|Im(c_(k,j))*A+Re(c_(k,j))*B| over L; and c. choosing the symbol timingĵ with the largest u_(j) by${\hat{j} = {\arg\left\{ {\max\limits_{j}\; u_{j}} \right\}}};$ (26.3)demodulating the complex digital decimated signal z_(k,ĵ) to generatethe binary sequence b(k)={{circumflex over (b)}₀, {circumflex over(b)}₁, {circumflex over (b)}₂, . . . , {circumflex over (b)}_(k−2),{circumflex over (b)}_(k−1), {circumflex over (b)}_(k), {circumflex over(b)}_(k+1), . . . } by: a. obtaining a first angle θ by Θ=C₂{circumflexover (Ω)}_(Δ)T−b_(k−1)·πhδ, wherein h is a demodulation index in theGFSK communication system${\delta = \frac{\int_{- T}^{0}{{p(t)}{\mathbb{d}t}}}{\int_{- \infty}^{\infty}{{p(t)}{\mathbb{d}t}}}},$ is a Gaussian function; b. obtaining a second angle Λ by$\Lambda = {{C_{3}\hat{\Omega_{\Delta}}T} + \left\{ \begin{matrix}{{{- \pi}\; h} + {2{\delta\pi}\; h}} & {{b_{k - 2} = 0},{b_{k - 1} = 1}} \\{{\pi\; h} - {2{\delta\pi}\; h}} & {{b_{k - 2} = 1},{{b_{k - 1} = 0};\mspace{14mu}{and}}} \\{{- \pi}\;{h \cdot b_{k - 1}}} & {otherwise}\end{matrix} \right.}$ c. obtaining b(k)={{circumflex over (b)}₀,{circumflex over (b)}₁, {circumflex over (b)}₂, . . . , {circumflex over(b)}_(k−2), {circumflex over (b)}_(k−1), {circumflex over (b)}_(k),{circumflex over (b)}_(k+1), . . . } by{circumflex over (b)} _(k) =sgn[Im(z _(k−1,ĵ) −z _(k,ĵ) ^(*) ·e ^(jΘ) +C₁ ·z _(k+1,ĵ) ·z _(k−1,ĵ) ^(*) ·e ^(jΛ))],  where C₁, C₂, C₃ areconstants, z_(k,ĵ)* is the conjugate complex number of z_(k,ĵ), and“sgn” represents taking sign of a quantity by “0” for a negative signand “1” for a positive sign and zero; (26.4) obtaining a starting bit{circumflex over (b)}_({circumflex over (k)}) of the binary sequenceb(k)={{circumflex over (b)}₀, {circumflex over (b)}₁, {circumflex over(b)}₂, . . . , {circumflex over (b)}_(k−2), {circumflex over (b)}_(k−1),{circumflex over (b)}_(k), {circumflex over (b)}_(k+1), . . . } by: a.responsive to the binary sequence b(k)={{circumflex over (b)}₀,{circumflex over (b)}₁, {circumflex over (b)}₂, . . . , {circumflex over(b)}_(k−2), {circumflex over (b)}_(k−1), {circumflex over (b)}_(k),{circumflex over (b)}_(k+1), . . . } and the preamble data t(k),obtaining a series of coefficient coff(k) by summation ofb(k+n)(xor)t(k) over M, wherein M is number of bits; and b. choosing thestarting bit {circumflex over (b)}_({circumflex over (k)}) of the binarysequence b(k)={{circumflex over (b)}₀, {circumflex over (b)}₁,{circumflex over (b)}₂, . . . , {circumflex over (b)}_(k−2), {circumflexover (b)}_(k−1), {circumflex over (b)}_(k), {circumflex over (b)}_(k+1),. . . } by min(coff(k)).